\input amstex
\documentstyle{amsppt}
\magnification=\magstep1

\pagewidth{6.5truein}
\pageheight{9truein}
\define\({\left(}
\define\){\right)}
\def\s#1#2{\medbreak
  \noindent{\bf#1.\enspace}{#2}\par
  \ifdim\lastskip<\medskipamount\removelastskip\penalty55\medskip\fi}
\def\t#1#2{\medbreak
  \centerline{{\bf#1.\enspace}{#2}}\par
  \ifdim\lastskip<\medskipamount\removelastskip\penalty55\medskip\fi}
\def\R{{\Bbb R}}
\nopagenumbers
\leftheadtext{Ph.D.~Algebra Exam, Spring 1995}
\topmatter
\title
\bf Ph.D. Comprehensive Examination\\
Algebra Section %PhD Algebra Exam
\endtitle
\author
\bf January 1995
\endauthor
\endtopmatter
%\head{\bf PhD Algebra Exam}\endhead
%\centerline{January 1995}%\endhead

\document
\t{Part I}{Do three (3) of these problems.}

\s{I.1}
{ If a subgroup $G$ of the symmetric group $S_n$ contains an odd
      permutation, then $|G|$ is even and exactly half the elements of $G$
      are odd permutations.}

\s{I.2}
{ Let $R$ be a commutative ring with no nonzero nilpotent elements
      (that is, $a^n=0$ implies $a=0$). If the polynomial $f(X)=a_0+a_1X+
      \ldots+a_mX^m$ in $R[X]$ is a zero-divisor (that is, $g(X)f(X)=0$
      for some nonzero polynomial $g(X)\in R[X]$), prove that there is an
      element $b\neq 0$ in $R$ such that $ba_0=ba_1=\ldots ba_m=0$.}

\s{I.3}
{ Let $V$ be a finite-dimensional vector space over a field $F$. An
      endomorphism $\phi$ of $V$ is called a {\it pseudoreflection\/} if
      $\phi-1$ has rank at most 1. Prove:}
\roster
      \item "a)" $\phi$ is a pseudoreflection precisely if there exists a basis
      of $V$ such that the matrix of $\phi$ has the form
      $$
      \left[\matrix
        \ast & \ast & \ast & \dots & \ast \\
        0 & 1 & 0 & \dots & 0 \\
        0 & 0 & 1 & \dots & 0 \\
        \vdots & \vdots & & \ddots &  \\
        0 & 0 & 0 & \dots & 1
      \endmatrix\right].
      $$
      \item "b)" Show that the Jordan canonical form of a pseudoreflection
$\phi$ is
      $$
      \left[\matrix
        1 & 1 & 0 & \dots & 0 \\
        0 & 1 & 0 & \dots & 0 \\
        0 & 0 & 1 & \dots & 0  \\
        \vdots & \vdots & & \ddots & \\
        0 & 0 & 0 & \dots & 1
      \endmatrix\right]\qquad\text{ or }\qquad
      \left[\matrix
        \ast & 0 & \dots & 0 \\
        0 & 1 & \dots & 0 \\
        \vdots & & \ddots &  \\
        0 & 0 & \dots & 1
      \endmatrix\right].
      $$
\endroster
 

\s{I.4}{ Let $F\supseteq K$ be an algebraic extension of fields and let $R$ be a
      subring of $F$ with $R\supseteq K$. Show that $R$ is a field.}

\vglue\baselineskip
\vfill


\t{Part II}{Do two (2) of these problems.}

\s{II.1}{ Let $G$ be a finite group and let $H$ be a proper subgroup of $G$.
      Show that $G$ is not the set-theoretic union of all conjugates of $H$.}

\s{II.2}{ Let $K$ be the splitting field over the rationals $\Bbb Q$ for the
polynomial $f(x)$. For each of the following examples, find the degree $[K:\Bbb
Q]$, determine the structure of the Galois group $G(K/\Bbb Q)$, describe its
action on the roots of $f(x)$ and identify the group.}
 \roster 
 \item "a)" $f(x) = x^4 -3$
 \item "b)" $f(x) = x^4 + x^2 - 6$
 \endroster

\s{II.3}{Let $G$ be a group of order $165=3\cdot 5\cdot 11$. Prove:}
 \roster
      \item "a)" $G$ has a normal Sylow 11-subgroup, say $C$.
      \item "b)" $G/C$ is cyclic. (HINT: Show that every group of order 15
            is cyclic.)
      \item "c)" $G$ has normal subgroups of orders 33 and 55.
  \endroster

\end{document}